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https://mathoverflow.net/questions/19282 | 20 | That's the question. Recall that a morphism $f\colon A\to B$ of commutative rings is integral if every element in $B$ is the root of a monic polynomial with coefficients in the image of $A$ and that $f$ is an epimorphism if and only if the multiplication map $$B\otimes\_A B\to B$$ is an isomorphism.
If we make the ad... | https://mathoverflow.net/users/1114 | Is every integral epimorphism of commutative rings surjective? | If I'm not mistaken, there is a counter-example. Have a look at Lazard's second counter-example in:
["Deux mechants contre-exemples" in Séminaire Samuel, Algèbre commutative, 2, 1967-1968](http://archive.numdam.org/ARCHIVE/SAC/SAC_1967-1968__2_/SAC_1967-1968__2__A8_0/SAC_1967-1968__2__A8_0.pdf).
For any field $k$, La... | 11 | https://mathoverflow.net/users/5417 | 21718 | 14,363 |
https://mathoverflow.net/questions/21717 | 7 | Background and motivation:
I am teaching the "covering space" section in an introductory algebraic topology course. I thought that, in the last five minutes of my last lecture, I might briefly sketch how to compute the "fundamental group of a field," primarily as a way of illustrating the analogy between Galois theory ... | https://mathoverflow.net/users/5094 | The etale fundamental group of a field | Essentially by definition, the etale fundamental group of a field $k$ is the group of $k$-automorphisms of a separable algebraic closure $k^{\operatorname{sep}}$ over $k$. If $k$ is perfect, then this is (well-defined up to an inner automorphism) $\operatorname{Aut}(\overline{k}/k)$. On the other hand, any $k$-automorp... | 9 | https://mathoverflow.net/users/1149 | 21719 | 14,364 |
https://mathoverflow.net/questions/21744 | 7 | Hello everyone, I am a newbie to the category theory. While reading some paper about infinite Galois theory, the 'anti-equivalence' of two categories showed up. Could anyone give me a 'good' explanation what this means? Thanks a lot.
| https://mathoverflow.net/users/3849 | What is an antiequivalence of two categories? | All it means that one of the categories is equivalent to the opposite of the other.
Wikipedia has informative pages on opposites of categories and equivalences
of categories:
<http://en.wikipedia.org/wiki/Opposite_%28category_theory%29> ,
<http://en.wikipedia.org/wiki/Equivalence_of_categories> .
| 16 | https://mathoverflow.net/users/4213 | 21746 | 14,378 |
https://mathoverflow.net/questions/15351 | 6 | Let $\Lambda$ be an $n$-dimensional lattice with basis $b\_1,\ldots,b\_n$. The problem of finding a "good" basis for $\Lambda$, or reducing a "bad" basis into a good one, is a very active area of research. Most basis reduction schemes try to optimize the norms of the basis vectors and their inner products. The goal is ... | https://mathoverflow.net/users/40739 | How to define a Voronoi reduced basis? | Excellent question. I don't know the answer and perhaps what I am suggesting is obvious. Nevertheless, I think it's on the right track.
Let $B$ be a basis for $\Lambda$ and let $P$ be the corresponding (origin centred) fundamental parallelepiped. That is, $P$ is the region given by $Bu$ where $u \in [-0.5,0.5]^n$. Cl... | 2 | https://mathoverflow.net/users/5378 | 21747 | 14,379 |
https://mathoverflow.net/questions/21742 | 7 | I'm interested in the following collection of questions: Let $S^n\_k = \sqcup\_k S^n$ be a disjoint union of $k$ distinct $n$-dimensional spheres. Write $Emb(S\_k^n, S^{n+2})$ for the space of embeddings of these spheres into $S^{n+2}$. Pick your favorite embedding $e: S\_k^n \to S^{n+2}$, and let $X\_e = S^{n+2} \setm... | https://mathoverflow.net/users/4649 | Knot complement diffeomorphism groups and embedding spaces | There is a locally-trivial fibre bundle
$$ Diff(S^{n+2}, L) \to Diff(S^{n+2}) \to Emb\_L(\sqcup\_k S^n, S^{n+2})$$
here $Emb\_L(\sqcup\_k S^n, S^{n+2})$ is the component of the link $L$ you're interested in the full embedding space $Emb(\sqcup\_k S^n,S^{n+2})$ and to simplify technicalities, assume $Diff(S^{n+2})$ ... | 8 | https://mathoverflow.net/users/1465 | 21760 | 14,384 |
https://mathoverflow.net/questions/21765 | 2 | Is it true that if a module has a free resolution of length $d$ then any of its submodule has a free resolution of length $\leq d$?
| https://mathoverflow.net/users/5292 | Projective dimension | No. A submodule of a free module need not have finite projective dimension.
As a simple example let $R=\mathbb{Z}/p^2\mathbb{Z}$. The free module $R$
has a submodule $p\mathbb{Z}/p^2\mathbb{Z}\cong\mathbb{Z}/p\mathbb{Z}$
which has no finite projective resolution.
| 8 | https://mathoverflow.net/users/4213 | 21766 | 14,389 |
https://mathoverflow.net/questions/21491 | 0 | I'm looking to efficiently zero-test "sparse integers", i.e. integers of the form $\sum C\_i \cdot A\_i^{X\_i}$ (where $A\_i, C\_i, X\_i$ are integers); equivalently test if a given (integer or rational) point is a zero of a sparse polynomial. For example, a randomised algorithm would be to compute the sum modulo a ran... | https://mathoverflow.net/users/5398 | abc-conjecture meets Catalan conjecture? | Regarding Paul's "first step question", I believe the answer may well be that the set of such M is finite. This follows under some coprimality hypothesis from the n-term abc conjecture of Browkin and Brzezinski [Math. Comp. 62 (1994), 931--939]. This states that, if we have
$a\_1, a\_2, \ldots, a\_n$ integers, for $n \... | 2 | https://mathoverflow.net/users/3533 | 21773 | 14,395 |
https://mathoverflow.net/questions/14384 | 15 | Warning: This one of those does-anyone-know-how-to-fix-this-vague-problem questions, and not an actual mathematics question at all.
If $X$ is a scheme of finite type over a finite field, then the zeta function $Z(X,t)$ lies in $1+t\mathbf{Z}[[t]]$. We can calculate the zeta function of a disjoint union by the formula... | https://mathoverflow.net/users/1114 | Can the failure of the multiplicativity of Euler factors at bad primes be corrected? | When I asked Niranjan Ramachandran this question a few days ago, he pointed out that you can indeed fix the problem if you work with integral models instead of varieties over $\mathbf{Q}$: Let $X$ be a scheme of finite type over $\mathbf{Z}$ and define the Euler factor $L\_p(X,s)$ to be $P\_p(X,p^{-s})$, where $P\_p(X,... | 4 | https://mathoverflow.net/users/1114 | 21777 | 14,398 |
https://mathoverflow.net/questions/21793 | 18 | An entertaining topological party trick that I have seen performed is to turn your pants inside-out while having your feet tied together by a piece of string. For a demonstration, check out this [video](http://www.youtube.com/watch?v=O70156EDGuY).
I have heard some testimonial evidence that it is also possible to tu... | https://mathoverflow.net/users/2233 | Turning pants inside-out (or backwards) while tied together | I think that the answer is no, by consideration of [linking numbers.](http://en.wikipedia.org/wiki/Linking_number)
First simplify the human body plus cord joining the ankles to a circle, and
assign it an orientation. Also assign an orientation to each pant cuff.
This can be done, e.g., so that each cuff has linking nu... | 25 | https://mathoverflow.net/users/1587 | 21798 | 14,403 |
https://mathoverflow.net/questions/21781 | 14 | I am trying to figure out when a closed, oriented manifold admits an orientation reversing diffeomorphism. My naive argument that the orientation cover should allow you to switch orientations is apparently wrong, since not every manifold admits such a diffeomorphism.
Can anyone give me some criteria for when such a m... | https://mathoverflow.net/users/3261 | Oriention-Reversing Diffeomorphisms of a Manifold | Such an endomorphism of $M$ gives an automorphism of the cohomology ring that acts by $-1$ on top cohomology. The cohomology ring of your example $M = {\mathbb C \mathbb P}^{2n}$ doesn't have such automorphisms.
| 18 | https://mathoverflow.net/users/391 | 21799 | 14,404 |
https://mathoverflow.net/questions/21800 | 11 | For reasons which are hard to articulate (due to they not being very clear in my mind), but having to do with the eprint [*From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group*](http://arxiv.org/abs/1002.1634) by Robert de Mello Koch and Sanjaye Ramgoolam, I have been wondering whether t... | https://mathoverflow.net/users/394 | Is there a notion of integration over the algebraic numbers? | You can definitely talk about integration on $\overline{\mathbb{Q}}$. The question is "with respect to what measure?" The reason integration theory works so well over a completion of $\mathbb{Q}$ is that such a field is, as an additive group, locally compact, and so possesses a Haar measure (a non-zero, translation inv... | 19 | https://mathoverflow.net/users/4351 | 21803 | 14,407 |
https://mathoverflow.net/questions/21785 | 7 | We know that for any topos E, and for any object A in E, the subobjects of A, Sub(A), form a Heyting lattice.
Does anybody know of any sort of modification of the definition of a topos that makes Sub(A) a different type of lattice? Could we get an incomplete lattice, or maybe a quantum lattice?
I'm curious because ... | https://mathoverflow.net/users/3664 | `Topos' with alternate subobject lattice? | Different types of categories lead to different types of [internal logics](http://ncatlab.org/nlab/show/internal+logic). Here is a very short list:
```
Regular Logic Regular Category
Coherent Logic Coherent Category
Geometric Logic Infinitary Coherent Category/Geometric Category
Fi... | 14 | https://mathoverflow.net/users/2000 | 21804 | 14,408 |
https://mathoverflow.net/questions/21783 | 1 | Can one use the real lie algebra so(3) to get knot polynomials? If so, do they have a skein relation (I presume they would, if they come from R-matrices in some standard way. If so, is the R-matrix written down anywhere?) I've only ever seen complex Lie algebras used to get knot polynomials, but I'm not sure if this is... | https://mathoverflow.net/users/492 | SO(3) knot polynomials | I don't see how taking a real form would *change* your knot invariants at all. Whatever it meant would be a number which would then not change when you base extend to the complex numbers.
From the quantum group perspective U\_q(so\_3) and U\_q(su\_2) mean the same thing (as far as I understand it), but for so\_3 the ... | 6 | https://mathoverflow.net/users/22 | 21807 | 14,410 |
https://mathoverflow.net/questions/21816 | 5 | I have a simple question. Let $C$ be a compact Riemann surface of genus, say $g >= 2$, to avoid silly cases.
I think it should be true, but I want to prove the following concretely:
"there exists a divisor $D$ on $C$ of degree $g-1$, that is non-special."
(For those who do not know what special divisors are: a d... | https://mathoverflow.net/users/3168 | Proving existence of non-special divisors of a given degree d on compact Riemann surfaces | Take $g+1$ general points $p\_1, \dots, p\_{g+1}$ on your curve. The divisor $p\_1+ \cdots +p\_g - p\_{g+1}$ is non-special. The proof is easy from the following lemma: if $D$ is a divisor such that $\mathrm{h}^0(D) > 0$, then $\mathrm{h}^0(D - p) = \mathrm{h}^0(D) - 1$ for all but finitely many points $p$. First you u... | 10 | https://mathoverflow.net/users/4790 | 21817 | 14,414 |
https://mathoverflow.net/questions/21796 | 4 | I was wondering, suppose I have a non-compact Kähler manifold $M$ and suppose that outside some compact subset $A\subset M$, there exists a smooth function $f:M\backslash A\longrightarrow\mathbb{R}$ such that $i\partial\bar{\partial}f>0$. Is it always possible for me to find a smooth function $h:M\longrightarrow\mathbb... | https://mathoverflow.net/users/4317 | Extension of strictly plurisubharmonic functions on a Kähler manifold | It is instructive to conisder the case of Kahler metrics invariant under torus action. In this case your question becomes a certain (nontivial) question on convex functions.
Recall first, that Kahler metrics on $(\mathbb C^\*)^n$ invariant under the action of $(S^1)^n$ have global potential that is given by a convex ... | 5 | https://mathoverflow.net/users/943 | 21824 | 14,418 |
https://mathoverflow.net/questions/21811 | 4 | In JP May's [Concise Course in Algebraic Topology](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf "Concise Course in Algebraic Topology"), on page 143 he says that the left- and right-multiplication-by-identity maps $\lambda:X\rightarrow X$ and $\rho:X\rightarrow X$ specify a map $X\vee X\rightarrow X$ th... | https://mathoverflow.net/users/303 | Can H-space multiplication always be straightened so that mult.-by-id. is the identity on the nose? | There are three possible definitions of an H-space, according to whether the "identity" element is a strict left and right identity, or only an identity up to basepoint-preserving homotopy, or just up to a non-basepoint-preserving homotopy. The three notions turn out to be equivalent, assuming the space is nice enough ... | 14 | https://mathoverflow.net/users/23571 | 21828 | 14,421 |
https://mathoverflow.net/questions/21833 | 5 | Background: By Chow's theorem, if a complex manifold can be embedded holomorphically into complex projective space, then this complex manifold must be algebraic.
Question: Suppose X is a compact complex manifold (not necessarily algebraic). Let $f:X--> {\mathbb{CP}}^n$ be a meromorphic map that is injective on its do... | https://mathoverflow.net/users/nan | Can a non-algebraic complex manifold be embedded meromorphically into projective space? | If $X$ is not compact there are loads of problems so I follow you in adding compactness as a condition.
Under the assumption of the question $X$ is bimeromorphic to a projective variety and hence by a result of Artin it is an algebraic space. Hence if you accept algebraic spaces as being algebraic the answer is yes. As... | 11 | https://mathoverflow.net/users/4008 | 21835 | 14,427 |
https://mathoverflow.net/questions/19210 | 21 | The question.
-------------
Let $(X, J)$ be a complex manifold and $u$ a holomorphic vector field, i.e. $L\_uJ = 0$. The holomorphicity of $u$ implies that the Lie derivative $L\_u$ on forms preserves the (p,q) decomposition and also that it commutes with $\bar{\partial}$. From this it follows that $u$ acts infinites... | https://mathoverflow.net/users/380 | Holomorphic vector fields acting on Dolbeault cohomology | Take a complex nilpotent or solvable group $G$ with the right action by a co-compact lattice $\Gamma$ and conisder the quotient $G/\Gamma$. On this quotient right-invariant $1$-forms give a subspace of $H^{1,0}$. The group $G$ is acting on $G/\Gamma$ on the left and if it would presrve all the $1$-forms, $G$ would be a... | 6 | https://mathoverflow.net/users/943 | 21840 | 14,429 |
https://mathoverflow.net/questions/21839 | 2 | I'm a newcomer to the realm of queueing theory, so please bear with me :)
I'd like to model web server traffic with a modified M/M/1 queue.
In the simple case we have two parameters - $\lambda$ for the arrival rate and $\mu$ for the departure (or service) rate.
If I understand correclty, the general way to get the ... | https://mathoverflow.net/users/4976 | "Induced" arrivals in an M/M/1 queue? | Just introduce extra states. The total description of a state will include the length of the queue and the stage of service for the current customer. For instance, if each initial service may result in the second stage service with probability $q$ and the departure rate for this second stage service is $\sigma$, then y... | 3 | https://mathoverflow.net/users/1131 | 21845 | 14,434 |
https://mathoverflow.net/questions/21852 | 3 | Hello to all,
I have been looking quite recently at the following theorem:
Let $X$ be a projective variety and $T$ a tilting object for $X$. If $A:=End(T)$ is the associated endomorphism algebra, then the functor
$RHom(T -): D^b(X) \rightarrow D^b(A)$ is in fact an equivalence. Now, this is proven (as in the claasica... | https://mathoverflow.net/users/4863 | Tensor product of sheaves and modules | What I've seen is a construction of a quasi-inverse for RHom(T,-), defined as $-\otimes\_A^L T$, as you wrote.
This last symbol should be interpreted as follows.
Given a left A-module M, we define a presheaf which with each U associates
$M\otimes^L\_A T(U)$.
Finally $M\otimes^L\_A T$ is defined as the sheafification of... | 3 | https://mathoverflow.net/users/3701 | 21856 | 14,440 |
https://mathoverflow.net/questions/21854 | 20 | I have a question about vector bundles on the algebraic surface $\mathbb{P}^1\times\mathbb{P}^1$. My motivation is the splitting theorem of Grothendieck, which says that every algebraic vector bundle $F$ on the projective line $\mathbb{P}^1$ is a direct sum of $r$ line bundles, where $r$ is the rank of $F$.
My questi... | https://mathoverflow.net/users/5395 | Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$ | The splitting theorem is most certainly false for vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$. In fact, the theory of vector bundles on quadric surfaces is probably as complicated as the theory of vector bundles on $\mathbb{P}^2$ (that is, very complicated).
Here is a simple example of an indecomposable rank 2... | 27 | https://mathoverflow.net/users/4790 | 21863 | 14,443 |
https://mathoverflow.net/questions/21859 | 2 | Hello.
Background
----------
Consider a weighted graph $G=(V,E,w)$. We are given a family of $k$ disjoint subsets of vertices $V\_1, \cdots, V\_k$.
A Steiner Forest is a forest that for each subset of vertices $V\_i$ connects all of the vertices in this subset with a tree.
Example: only one subset of vertices $... | https://mathoverflow.net/users/5466 | An approximate algorithm for finding Steiner Forest in a graph. | There is a 2-approximation algorithm, see e.g.
>
> A General Approximation Technique For Constrained Forest Problems, Michel Goemans, David P. Williamson, SIAM Journal on Computing 1992.
>
>
>
For special kind of graphs, better bounds can be obtained: for planar graphs there is a PTAS,
>
> Approximation S... | 4 | https://mathoverflow.net/users/4248 | 21865 | 14,445 |
https://mathoverflow.net/questions/21857 | 11 | This semester I am attending a reading seminar on non-archimedean analytic geometry (a subject I know nothing about), roughly following the [notes of Conrad](http://math.arizona.edu/~swc/aws/07/speakers/index.html).
Reading Conrad's notes (and e.g. those of Bosch) it struck me that the prime spectrum of affinoid alg... | https://mathoverflow.net/users/nan | Why is the prime spectrum not useful in non-archimedean analytic geometry? | I am surprised that Brian got to this one first without making what I thought was another obvious comment: affinoids are Jacobson rings! A function which is zero at all points of an affinoid rigid space corresponds to an element of your affinoid algebra which is in all maximal ideals and hence (by Jacobson-ness) is nil... | 13 | https://mathoverflow.net/users/1384 | 21876 | 14,451 |
https://mathoverflow.net/questions/21879 | 3 | This question stems from Dick Lipton's [recent blog post](http://rjlipton.wordpress.com/2010/04/12/socks-shoes-and-the-axiom-of-choice/) on the Axiom of Choice. I asked there but got no takers. I promise I'm not an inept Googler, but I couldn't find a satisfactory answer. I suspect universal in this context means compu... | https://mathoverflow.net/users/175 | What is a universal function? | In that argument, he just means that g is defined on all the two-element subsets that may arise in the argument, that is, for a given family F of four-element sets, g(A) should be defined on any two-element set A that is a subset of a four-element set in F.
The reason he needs to assume that is that he cannot allow ... | 5 | https://mathoverflow.net/users/1946 | 21884 | 14,454 |
https://mathoverflow.net/questions/3330 | 19 | I've been thinking a lot lately about random permutations. It's well-known that the mean and variance of the number of cycles of a permutation chosen uniformly at random from Sn are both asymptotically log n, and the distribution is asymptotically normal.
I want to know what a typical permutation of [n] with k(n) cyc... | https://mathoverflow.net/users/143 | How can I generate random permutations of [n] with k cycles, where k is much larger than log n? | Youll find the answer on page 38 of my lecture notes East Side, West Side, which are a free download from my web site. It's a complete, short Maple program. The problem was originally solved in 1978 in Combinatorial Algorithms, by Albert Nijenhuis and myself.
Herb Wilf
| 21 | https://mathoverflow.net/users/5477 | 21896 | 14,461 |
https://mathoverflow.net/questions/21745 | 11 | I would like to know which is the real difference between the Recursive topos (in the sense of Mulry) and the Effective topos (in the sense of Hyland). Especially what is related to recursive functions. Do they have the same semantic power?
I will be gratefull with some hints about texts related to this.
Thanks in ad... | https://mathoverflow.net/users/3338 | The difference between the Recursive and the Effective topos. | Your question is a bit unclear, but an obvious difference between these two toposes is that the Recursive Topos is a topos of sheaves, hence cocomplete, wherease the Effective topos only has finite (non-trivial) coproducts. For example, the natural numbers object in the Effective topos is *not* a countable coproduct of... | 14 | https://mathoverflow.net/users/1176 | 21898 | 14,463 |
https://mathoverflow.net/questions/21775 | 5 | My collaborators and I are studying certain rigidity properties of hyperbolic toral automorphisms.
These are given by integral matrices A with determinant 1 and without eigenvalues on the unit circle.
We obtain a result under two additional assumptions
1) Characteristic polynomial of the matrix A is irreducible
... | https://mathoverflow.net/users/2029 | Spectrum of a generic integral matrix. | Yes, a generic integer matrix has no more than two eigenvalues of the same norm. More precisely, I will show that matrices with more than two eigenvalues of the same norm lie on a algebraic hypersurface in $\mathrm{Mat}\_{n \times n}(\mathbb{R})$. Hence, the number of such matrices with integer entries of size $\leq N$... | 6 | https://mathoverflow.net/users/297 | 21906 | 14,467 |
https://mathoverflow.net/questions/21881 | 72 | I am a TA for a multivariable calculus class this semester. I have also TA'd this course a few times in the past. Every time I teach this course, I am never quite sure how I should present curl and divergence. This course follows Stewart's book and does not use differential forms; we only deal with vector fields (in $\... | https://mathoverflow.net/users/83 | How should one present curl and divergence in an undergraduate multivariable calculus class? | To me, the explanation for the appearance of div, grad and curl in physical equations is in their invariance properties.
Physics undergrads are taught (aren't they?) Galileo's principle that physical laws should be invariant under inertial coordinate changes. So take a first-order differential operator $D$, mapping ... | 50 | https://mathoverflow.net/users/2356 | 21908 | 14,469 |
https://mathoverflow.net/questions/21890 | 6 | Background and Motivation
-------------------------
Local Class Field Theory says that abelian extensions of a finite extension $K/\mathbb{Q}\_p$ are parametrized by the open subgroups of finite index in $K^\times$. The correspondence takes an abelian extension $L/K$ and sends it to $N\_{L/K}(L^\times)$, and this cor... | https://mathoverflow.net/users/5473 | The norm of a non-Galois extension of local fields | Yup, it is. This is Theorem III.3.5 (norm limitation theorem) of Milne's Class field theory notes (available [here](http://jmilne.org/math/CourseNotes/CFT.pdf)). The global analogue is Theorem VIII.4.8.
| 4 | https://mathoverflow.net/users/1021 | 21909 | 14,470 |
https://mathoverflow.net/questions/21901 | 5 | In a review of a book by Ferdinand Gonseth in the Spring-Summer 2006 (Volume XI, Issue 1) of the HOPOS Newsletter it is said that Gonseth was the
"successor on Jerome's (sic) Franel's chair for Mathematics in French language at the ETH"
1) Was this a chair that Franel held or was it a chair Franel endowed? If the ... | https://mathoverflow.net/users/4111 | Jerome Franel's Chair at ETH | Franel was a number theorist who gave introductory calculus lectures in French
at the ETH. Maybe the ETH offered courses in different languages, because of its location in a multilingual country. Does anybody know? Anyway, for a little more information, see [this excerpt](http://books.google.com/books?id=OuHrR_6WEKsC&l... | 6 | https://mathoverflow.net/users/1587 | 21914 | 14,474 |
https://mathoverflow.net/questions/21877 | 4 | Given $n,k\in\mathbb{N}$ where $k\leq n$, I want to compute the minimum subset of the set of partitions of $N$={$1,\ldots,n$}, satisfying these properties:
1. Each block of every partition has at most $k$ elements.
2. Every pair of elements of $N$ is in the same block in exactly one partition.
Anyone has a clue?
| https://mathoverflow.net/users/961 | Minimum cover of partitions of a set | (Edit: Sorry, my original restatement was incorrect.)
This problem is equivalent to decomposition a complete graph $K\_n$ into a collection of cliques $C:=\{K\_s\}$ where each $s \leq k$, such that $C$ can be resolved (i.e. partitioned) into a set of resolution classes $\mathcal{P}$ (the vertices of the graphs within... | 7 | https://mathoverflow.net/users/2264 | 21921 | 14,478 |
https://mathoverflow.net/questions/21899 | 28 | I've tried in vain to find a definition of an algebra over a *noncommutative* ring. Does this algebraic structure not exist? In particular, does the following definition from <http://en.wikipedia.org/wiki/Algebra_(ring_theory)> make sense for noncommutative $R$?
>
> Let $R$ be a commutative ring. An algebra is an $... | https://mathoverflow.net/users/1291 | Definition of an algebra over a noncommutative ring | The commutative notion of an (associative or not) algebra $A$ over a commutative ring $R$ has two natural generalization to the noncommutative setup, but the one you list with defined **left** $R$-linearity in both arguments is neither of them; in particular your multiplication does not necessarily induce a map from th... | 23 | https://mathoverflow.net/users/35833 | 21927 | 14,481 |
https://mathoverflow.net/questions/21688 | 7 | How does one study Krull dimension of some I-adic completion of a ring or, more generally, a module? I know that in case of Noetherian local ring Krull dimension of its completion equals Krull dimension of the ring, but what can we say in general case?
| https://mathoverflow.net/users/5432 | Krull dimension of a completion | For a Noetherian ring R, the Krull dimension of its $I$-adic completion, $\hat{R}$ is given by $\sup h(J)$, where $J$ ranges over all maximal ideals of $R$ containing $I$ and $h(J)$ is the [height](http://en.wikipedia.org/wiki/Height_(ring_theory)) of $J$. Therefore $\dim \hat R\le \dim R$ with equality only when $I\su... | 13 | https://mathoverflow.net/users/2384 | 21930 | 14,483 |
https://mathoverflow.net/questions/21929 | 43 | We're all use to seeing differential operators of the form $\frac{d}{dx}^n$ where $n\in\mathbb{Z}$. But it has come to my attention that this generalises to all complex numbers, forming a field called fractional calculus which apparently even has applications in physics!
These derivatives are defined as fractional it... | https://mathoverflow.net/users/5486 | What is the actual meaning of a fractional derivative? | I understand where Ryan's coming from, though I think the question of how to interpret fractional calculus is still a reasonable one. I found this paper to be pretty neat, though I have no idea if there are any better interpretations out there.
<http://people.tuke.sk/igor.podlubny/pspdf/pifcaa_r.pdf>
| 22 | https://mathoverflow.net/users/1916 | 21933 | 14,485 |
https://mathoverflow.net/questions/21935 | 8 | Requesting: a good reference for formal manipulation of limits of diagrams, with respect to maps of index diagrams.
As an example, consider the following result, for some "nice enough" category C (say, Top), there is a natural isomorphism
$$(A \times\_B C) \times\_C D \cong A \times\_B D.$$
This is a nice, intuitivel... | https://mathoverflow.net/users/2532 | How to formally -- and cleanly -- express relationships of limits of diagrams? | As was mentioned in the comments, such general isomorphisms can be reduced by Yoneda to the case of sets, and in your example $((a,c),d) \mapsto (a,d)$ is an isomorphism, simply because $c$ is already determined by $d$. As for me, I always manipulate arrows without caring about single-use names *and* use Yoneda. This w... | 3 | https://mathoverflow.net/users/2841 | 21948 | 14,491 |
https://mathoverflow.net/questions/21940 | 5 | Let $I$ be a normal ideal on $P\_{\kappa} (\lambda)$. Let $V$ denote our ground model. Now we force with the $I$-positive sets, and if $G$ is the resulting generic filter, it can be shown that $G$ is a $V$-ultrafilter on $P\_{\kappa} (\lambda)$ extending the dual filter of $I$, which is normal if $I$ is normal.
We ca... | https://mathoverflow.net/users/4753 | The closure of a generic ultrapower | If your ideal is normal, fine, precipitous and has the disjointing property (a consequence of saturation), then the answer is yes. As you likely know, you need more assumptions than you had stated, just in order to know that the ultrapower is well-founded. The difference in closure for the ultrapower that you mentioned... | 5 | https://mathoverflow.net/users/1946 | 21963 | 14,500 |
https://mathoverflow.net/questions/21959 | 10 | Let G be a Lie group and M a smooth manifold. Suppose that P is a principal G-bundle over M. Then by Yoneda, this corresponds to a smooth map $p:M \to [G]$, where $[G]$ is the differentiable stack associated to G. If P is equipped with a connection, how does this fit in this picture?
I was thinking of using the fact ... | https://mathoverflow.net/users/4528 | Connections on principal bundles via stacks? | The only thing which has to be replaced in the representation of a principal bundle as a suitable class of a maps $M\to [\*/G]$ in order to introduce a flat connection is to replace $M$ by its fundamental 1-groupoid $\Pi\_1(M)$, or, if the connection is not flat, by the thin homotopy version $P\_1(M)$ of it, cf. [nlab:... | 5 | https://mathoverflow.net/users/35833 | 21967 | 14,504 |
https://mathoverflow.net/questions/21820 | 3 | Let $A \subseteq \mathcal B(\mathcal H)$ be a unital C\*-algebra in its universal representation. The GNS representation $\pi\_\mu\colon A \rightarrow \mathcal B(\mathcal H\_\mu)$ with base state $\mu$ extends uniquely to a normal $\ast$-homomorphism $\pi\_\mu''\colon A'' \to \mathcal B(\mathcal H\_\mu)$. Since $A''$ i... | https://mathoverflow.net/users/2206 | Kernel projections in the universal representation. | So, it seems like the new question is: Is $\ker\pi\_\mu \subseteq A$ dense in $\ker\pi\_\mu'' \subseteq A''$. As we're talking about the universal representation, $A''=A^{\*\*}$, the bidual of A.
So, suppose that $\mu$ is a *faithful* state on A, so $\ker\pi\_\mu=\{0\}$. I don't think it's necessary that $\ker \pi\_\... | 3 | https://mathoverflow.net/users/406 | 21970 | 14,507 |
https://mathoverflow.net/questions/21916 | 0 | I am fairly new at linear programming/optimization and am currently working on implementing a linear program that is stated like this:
max $\sum\_{i=1}^{k}{p(\vec \alpha \cdot \vec c\_i)}$
$s.t. $
$|\alpha\_j| \le 1$
Where p(x) = 2x if x < 0, x otherwise, and $ \vec c$ is a constant
The p(x) function is what... | https://mathoverflow.net/users/5483 | Linear programming piecewise linear objective | Sorry for making you wait 14 hours unnecessarily but you are partially guilty yourself: if you posted a correct and full version of the question from the beginning, you would get the answer in 5 minutes. Keep it in mind when you ask a question on a public forum next time.
Your problem is equivalent to maximizing the ... | 3 | https://mathoverflow.net/users/1131 | 21974 | 14,510 |
https://mathoverflow.net/questions/21960 | 1 | Suppose to have a Lie algebra L with a reductive lie subalgebra G. Let l an element of L such that [l,g] is in G for every g in G, is it true that l is an element of G?if not, there are some restriction on G that makes it true?
| https://mathoverflow.net/users/4821 | reductive Lie subalgebra | Sorry, this started as a comment, but got too long.
If $G$ is semisimple, then every derivation of $G$ is inner, so
that the normalizer $N\_L(G)=C\_L(G)+G$ where $C\_L(G)$
is the centralizer. In this situtaion Michele's condition holds
if and only if the centralizer is trivial.
However the case where $G$ is reducti... | 4 | https://mathoverflow.net/users/4213 | 21978 | 14,512 |
https://mathoverflow.net/questions/21947 | 17 | In [this post](https://mathoverflow.net/questions/21745/the-difference-between-the-recursive-and-the-effective-topos) about the difference between the recursive and effective topos, Andrej Bauer said:
>
> If you are looking for a deeper explanation, then perhaps it is fair to say that the Recursive Topos models com... | https://mathoverflow.net/users/1610 | Why is Kleene's notion of computability better than Banach-Mazur's? | This answer requires a bit of background.
**Definition 1:** a *numbered set* $(X,\nu\_X)$ is a set $X$ together with a partial surjection $\nu\_X : \mathbb{N} \to X$, called a *numbering* of $X$. When $\nu\_X(n) = x$ we say that $n$ is a *code* for $x$.
Numbered sets are a generalization of Gödel codes. Some typica... | 23 | https://mathoverflow.net/users/1176 | 21979 | 14,513 |
https://mathoverflow.net/questions/21954 | 8 | Can someone please tell me where I can find a citeable reference for the following result:
Call the metric space $(X, d)$ "locally complete" if for every $x \in X$ there a neighbourhood of $x$ which is complete under $d$.
If $(X,d)$ is locally complete and separable then there exists a metric $d'$ on $X$ such that ... | https://mathoverflow.net/users/3676 | Locally complete space is topologically equivalent to a complete space | For a reference: [this paper](http://www.chimiefs.ulg.ac.be/SRSL/newSRSL/modules/FCKeditor/upload/File/73_1/Bella%20TSIRULNIKOV%20p%209-19.pdf) has a reference [30] that has a proof. The author cites your result and refers to it. I don't have access to these papers, so I cannot verify exactly.
| 3 | https://mathoverflow.net/users/2060 | 21983 | 14,516 |
https://mathoverflow.net/questions/21977 | 1 | Can any one give an example of a 3-fold X which contains an embedded ample divisor $D \cong CP^2$
with normal bundle $O(3)$ in X?
| https://mathoverflow.net/users/5259 | Looking for a 3-fold with some property? | Let me give an attempt of a proof of the fact that such example does not exist.
Proof. Suppose $X$ is such a $3$-fold and let $L$ be the line bundle corresponding to the divisor $\mathbb CP^2$. First we will prove that $Pic(X)=\mathbb Z$ and then will get a contradiction.
Notice that $L$ has a lot of sections. In... | 4 | https://mathoverflow.net/users/943 | 21984 | 14,517 |
https://mathoverflow.net/questions/21903 | 24 | I'm curious if the term localization in ring theory comes from algebraic geometry or not. The connection between localization and "looking locally about a point" seems like it should be the source for the notion of localization. It seems plausible, but it seems like we would have had to wait until Zariski defined the Z... | https://mathoverflow.net/users/1353 | Origin of the term "localization" for the localization of a ring | I'm looking at the paper "On the theory of local rings" by Chevalley (Annals of Math. 44 (1943)). In this paper he explains how to localize at a multiplicative set $S$ of non-zero divisors, and calls this the *ring of quotients* of the set $S$.
There is no question that Chevalley was motivated by algebraic geometry.... | 14 | https://mathoverflow.net/users/2874 | 21987 | 14,519 |
https://mathoverflow.net/questions/21961 | 13 | In the Euclidean plane, is the circle the only simple closed curve that has an axis of symmetry in every
direction?
| https://mathoverflow.net/users/4423 | Can the circle be characterized by the following property? | A slightly different argument is as follows. Choose two symmetries $\sigma,\tau$
with axes
intersecting at a point $P$ and forming an angle of $2\pi \lambda$ with $\lambda$ irrational.
The composition $\rho=\sigma\circ \tau$ is then a rotation of infinite order generating
a dense subgroup of the group of all rotations ... | 18 | https://mathoverflow.net/users/4556 | 21989 | 14,520 |
https://mathoverflow.net/questions/21939 | 31 | So sometime ago in my math education I discovered that many mathematicians were interested in moduli problems. Not long after I got the sense that when mathematicians ran across a non compact moduli they would really like to compactify it.
My question is, why are people so eager to compactify things? I know compactn... | https://mathoverflow.net/users/7 | What can you do with a compact moduli space? | The answers here are all excellent examples of things that can only be proved once a moduli space is compactified. I would like to add a perhaps more basic reason for compactifying moduli spaces, involving something simpler than theoretical applications such as defining enumerative invariants. The moral is the followin... | 19 | https://mathoverflow.net/users/380 | 21997 | 14,525 |
https://mathoverflow.net/questions/21998 | 4 | Hello, I'm interested in the case when all varieties are projective threefolds over the complex numbers.
Start with a flopping contraction $f:Y \rightarrow X$, with corresponding flop $f^+: Y^+ \rightarrow X$. It has been [proved](http://www.tombridgeland.staff.shef.ac.uk/papers/flop.pdf) that there then exists an eq... | https://mathoverflow.net/users/3701 | Is there a nice way to characterise the derived equivalence induced by a flop? | As always, it depends on what you think "explicitly" means. It's a Fourier-Mukai transform; see, for example, [Van den Bergh and Hille's expository article](http://alpha.uhasselt.be/Research/Algebra/Publications/hille.pdf). It can also be explained in terms of so-called non-commutative crepant resolutions, [see Van den... | 2 | https://mathoverflow.net/users/460 | 22000 | 14,527 |
https://mathoverflow.net/questions/22001 | 6 | I am teaching a graduate "classical" course on modular forms. I try to achieve the most elementary level for presenting modular polynomials. Serge Lang's "Elliptic functions" cover the topic quite well, except for some inconsistence (from my point of view) in using left/right cosets for actions of the modular group on ... | https://mathoverflow.net/users/4953 | References for modular polynomials | I'm not quite sure what exactly you're asking for, but you might find the third part of David Cox's *Primes of the form x^2 + n y^2* useful for an elementary approach to modular polynomials.
| 4 | https://mathoverflow.net/users/422 | 22005 | 14,530 |
https://mathoverflow.net/questions/22009 | 8 | I'm looking for an anecdote about a mathematician who studied random walks. I'm attempting to write an article and hope to include the story (but only if I can get the details correct). I'll try to do my best describing it in hopes someone else has heard it and knows a name or the full story.
A mathematician was walk... | https://mathoverflow.net/users/3737 | Random Walk anecdote. | The anecdote is about Polya, and it is in his contribution, Two incidents, to the book, Scientists at Work: Festschrift in Honour of Herman Wold, edited by T Dalenius, G Karlsson, and S Malmquist, published in Sweden in 1970. It was recently quoted on page 229 of David A Levin and Yuval Peres, Polya's Theorem on random... | 12 | https://mathoverflow.net/users/3684 | 22010 | 14,533 |
https://mathoverflow.net/questions/22012 | 15 | I have some understanding that vector bundles provide a basic, familiar example of what I should call a stack. Namely, consider the functor $Vect$ that assigns to a space $X$ the *set* of isomorphism classes of vector bundles on $X$. This functor isn't local, in the sense that the isoclass of a vector bundle isn't dete... | https://mathoverflow.net/users/361 | K-Theory and the Stack of Vector Bundles | [Read this](http://ncatlab.org/nlab/show/moduli+space#because)
| 7 | https://mathoverflow.net/users/83 | 22018 | 14,535 |
https://mathoverflow.net/questions/17565 | 9 | Quite a long ago, I tried to work out explicitly the content of the Newlander-Nirenberg theorem. My aim was trying to understand wether a direct proof could work in the simplest possible case, namely that of surfaces. The result is that the most explicit statement I could get is a PDE I was not able to solve.
Assume ... | https://mathoverflow.net/users/828 | Newlander-Nirenberg for surfaces | I'm not sure if the following is elementary enough, but it does only use standard PDE machinery (plus some basic Riemannian geometry). It's also nice in that it suggests an approach to proving the uniformization theorem (via metrics of constant curvature).
Say you have an almost-complex structure on the unit disk. Yo... | 8 | https://mathoverflow.net/users/5499 | 22019 | 14,536 |
https://mathoverflow.net/questions/22017 | 11 | Let $S\_{n}$ denote the permutation group on $n$ letters and $G\subset S\_{n}$ a transitive subgroup. The inclusion of $G$ in $S\_{n}$ defines an action of $G$ on $\mathbb{C}^{n}$. By finding a generating set of invariant polynomials and relations among these polynomials we may realize the quotient space $\mathbb{C}^{n... | https://mathoverflow.net/users/5124 | When Are Quotients Complete Intersections? | This paper of Kac and Watanabe may be of interest:
Kac, Victor; Watanabe, Keiichi, *Finite linear groups whose ring of invariants is a complete intersection*
<http://www.ams.org/mathscinet-getitem?mr=640951>
From the review:
"The following theorems are proved: Let $k$ be a field and $G$ be a finite subgroup of $\... | 14 | https://mathoverflow.net/users/321 | 22020 | 14,537 |
https://mathoverflow.net/questions/22014 | 5 | I was reading a proof that used the following result
Let $C$ be a hyperelliptic of genus $\ge 3$ and $\tau \colon C \to C$ the hyperelliptic involution. If $D$ is an effective divisor of degree $g-1$ such that $h^0(D)>1$ then $D = x + \tau(x) + D'$ where $D'$ is an effective divisor.
My question is, how is this res... | https://mathoverflow.net/users/7 | Special divisors on hyperelliptic curves | Suppose that $D=x\_1+x\_2+\cdots+x\_{g-1}$. We may assume that $\tau(x\_i)\neq x\_j$
for all $i\neq j$. Now, assume that $D'=y\_1+\cdots+y\_{g-1}$ is an element of
$|K-D|$. Then $x\_1+x\_2+\cdots+x\_{g-1}+y\_1+\cdots+y\_{g-1}$ is an element of $|K|$
but we know that any such element is of the form
$z\_1+\tau(z\_1)+\cdo... | 12 | https://mathoverflow.net/users/4008 | 22022 | 14,539 |
https://mathoverflow.net/questions/22013 | 4 | Given a completely metrizable space, say that it has property X if it can be embedded in some metric space such that its image is not closed. For example, the real line R can be embedded, topologically, in itself as (0,1) which is not closed. A compact space such as S^1, however, clearly cannot be embedded in any metri... | https://mathoverflow.net/users/4336 | Topological embeddings of non-compact, complete metric spaces | The answer to 2 is yes. Let $(X,d)$ be complete but not compact, then for some $r>0$ it has a countable r-separated subset $S=\{s\_i\}:i=1,2,\dots$. Let $d'$ be the maximal metric on $X$ satisfying $d'\le d$ and $d'(s\_i,s\_j)\le r/\min(i,j)$ for all $i,j$. Then $(X,d')$ is homeomorphic to $(X,d)$ - in fact, the identi... | 7 | https://mathoverflow.net/users/4354 | 22031 | 14,546 |
https://mathoverflow.net/questions/21589 | 9 | Even searching for " ['number of trees' leaves](https://oeis.org/search?q=%22number+of+trees%22+leaves) " didn't reveal what I am looking for: an approach for calculating the (approximate) number of trees with exactly n nodes and m leaves. Any hints from MO?
| https://mathoverflow.net/users/2672 | Number of trees with n nodes and m leaves | The answer to this (very natural) question depends on your notion of "tree" (e.g. free, rooted) and the equivalence relation you employ (e.g. labelled, unlabelled). I haven't gone into the nitty-gritty details of all these results, but here's what I've found so far. There's likely published results I haven't found yet,... | 11 | https://mathoverflow.net/users/2264 | 22035 | 14,548 |
https://mathoverflow.net/questions/22028 | 4 | Given two $n-$dimensional convex compact sets $A,B$, we define $d(A,B)$ as $\log({\mathrm{Vol}}(\alpha\_2(A)))-\log(\mathrm{Vol}(\alpha\_1(A)))$ where $\alpha\_1,\alpha\_2$ are two affine bijections such that $\alpha\_1(A)\subset B\subset\alpha\_2(A)$ and such that the ratio $\mathrm{Vol}(\alpha\_2(A))/\mathrm{Vol}(\al... | https://mathoverflow.net/users/4556 | Diameter of a metric on orbits under affine bijections of $n-$dimensional convex compact sets | I assume you also want your compact sets to have non-empty interior, hence positive volume.
The literature mostly deals with the related [Banach-Mazur](http://en.wikipedia.org/wiki/Banach-Mazur_compactum) metric $d\_{BM}(A,B)$, in which it is assumed that $\alpha\_1(A)$ and $\alpha\_2(A)$ are homothetic, so $d\_{BM}(... | 3 | https://mathoverflow.net/users/1044 | 22042 | 14,551 |
https://mathoverflow.net/questions/22039 | 5 | Let $S$ be a K3 surface over $\mathbb{C}$, that is, $S$ is a simply connected compact smooth complex surface whose canonical bundle is trivial. I recall reading somewhere that any two such surfaces are diffeomorphic, however I can't for the life of me remember where, or how the proof goes.
Does anybody know a good r... | https://mathoverflow.net/users/5101 | Are any two K3 surfaces over C diffeomorphic? | I think this was first proved by Kodaira. See [On the structure of complex analytic surfaces, 1](http://www.jstor.org/pss/2373157). There Kodaira proves that any K3 surface is a deformation of a non-singular quartic surface in $\mathbb{CP}^3$. In particular, they are all diffeomorphic.
| 15 | https://mathoverflow.net/users/380 | 22043 | 14,552 |
https://mathoverflow.net/questions/22036 | 4 | Do pullbacks exist in the category of sets and partial functions?
Are the 'the same' as they are in Sets? That is, given two partial functions $f : A \to C$ and $g : B \to C$, is the pullback given by $\{ (a,b) \in A\times B ~|~ f(a)=g(b) \}$?
If not, what is a simple description of the pullback?
| https://mathoverflow.net/users/2620 | Pullbacks in Category of Sets and Partial Functions | Pullbacks exists but are not what you describe.
The answer is as follows:
The category $\mathcal{C}$ of sets and partial functions is equivalent to the category of based sets and based functions, by sending the set $A$ to $A$ disjoint union a base point $\*$ and sending $f$ to the obvious based function which sends... | 7 | https://mathoverflow.net/users/4500 | 22046 | 14,555 |
https://mathoverflow.net/questions/22040 | 16 | Let $X$ be a connected affine variety over an algebraically closed field $k$, and let $X \subset Y$ be a compactification, by which I mean $Y$ is a proper variety (or projective if you prefer), and $X$ is embedded as an open dense subset.
I am guessing that it is not always the case that $Y\setminus X$ is a divisor,... | https://mathoverflow.net/users/5101 | Is the complement of an affine variety always a divisor? | It it true for any $Y$: see Corollaire 21.12.7 of EGAIV.
| 18 | https://mathoverflow.net/users/4790 | 22052 | 14,560 |
https://mathoverflow.net/questions/21703 | 10 | Greg Muller, in a post called [Rational Homotopy Theory](http://cornellmath.wordpress.com/2008/04/27/rational-homotopy-theory/) on the blog "The Everything Seminar" wrote
"I tend to think of homotopy theory a little bit like ‘The One That Got Away’ from mathematics as a whole. Its full of wistful fantasies about how... | https://mathoverflow.net/users/4692 | What would be the ramifications of homotopy theory being as easy as homology theory? | Homology groups and homotopy groups are two sides of the same story. Homotopy groups tell us all the ways we can have a map Sn → X, and in particular describe all ways we can attach a new cell to our space. On the other side, the homology groups of a space change in a very understandable way each time we attach a new c... | 12 | https://mathoverflow.net/users/360 | 22057 | 14,564 |
https://mathoverflow.net/questions/22050 | 9 | Is there a common name for the complement $\widehat{X} \setminus X$ of a metric space $X$ in its metric completion $\widehat{X}$? Since $X$ is not necessarily open in $\widehat{X}$, the term *boundary* is out of the question (without additional qualifiers). *Metric remainder* seems appropriate but I did not find it in ... | https://mathoverflow.net/users/2000 | Is there a common name for the complement of a metric space in its completion? | Remainder. I agree with that. But I don't find it on-line. Maybe "remainder" is primarily used for $\beta X \setminus X$ ? But it should be OK in your setting if you say the first time you use it: "the remainder of $X$ in its completion" or something.
| 4 | https://mathoverflow.net/users/454 | 22059 | 14,566 |
https://mathoverflow.net/questions/22080 | 17 | It is well-known that if $X$ is an integral scheme, then there is an isomorphism $CaCl(X)\to Pic(X)$ taking $[D]$ to $[\mathcal{O}\_X(D)]$. Does anyone know any simple examples where the above map fails to be surjective, i.e., a line bundle on a scheme $X$, not isomorphic to $\mathcal{O}\_X(D)$ for any Cartier divisor ... | https://mathoverflow.net/users/3996 | Line bundles vs. Cartier divisors on a non-integral scheme | An example is given in this [note](http://webs.uvigo.es/martapr/Investigacion/Divisors.pdf) (it was credited to Kleiman).
| 12 | https://mathoverflow.net/users/2083 | 22085 | 14,581 |
https://mathoverflow.net/questions/22062 | 12 | A *polyhedron* is the intersection of a finite collection of halfspaces. These halfspaces are not assumed to be *linear*, i.e. their bounding hyperplanes are not assumed to contain the origin. The *support* Supp(M) of a collection M of polyhedra is the union of the polyhedra in M.
I can prove the following theorem:
*... | https://mathoverflow.net/users/5519 | To what extent is convexity a local property? | This is known as Tietze theorem: if $A$ is an open connected set such that for every boundary point there is a locally supporting hyperplane, then $A$ is convex. I don't know what is the standard reference, internet search gave me the following one:
F.A.Valentine. Convex sets. McGraw-Hill, New York, 1964, pp. 51-53.
... | 19 | https://mathoverflow.net/users/4354 | 22091 | 14,583 |
https://mathoverflow.net/questions/22087 | 9 | In two different books I found these two related statements.
* The book by Jost defines a ``locally symmetric space" as one for which the curvature tensor is constant and which is geodesically complete.
* Andreas' book attributes to Cartan a theorem that a space is locally symmetric if and only if its curvature tenso... | https://mathoverflow.net/users/2678 | Constant curvature manifolds | It is difficult to reconcile your first two statements, because they are actually wrong as written!
A riemannian manifold is *locally symmetric* if and only if the Riemann curvature tensor is parallel with respect to the Levi-Civita connection. This condition was studied by Élie Cartan, who classified them using his ... | 11 | https://mathoverflow.net/users/394 | 22092 | 14,584 |
https://mathoverflow.net/questions/22032 | 62 | I read the article in wikipedia, but I didn't find it totally illuminating. As far as I've understood, essentially you have a morphism (in some probably geometrical category) $Y \rightarrow X$, where you interpret $Y$ as being the "disjoint union" of some "covering" (possibly in the Grothendieck topology sense) of $X$,... | https://mathoverflow.net/users/4721 | What is descent theory? | Suppose we are given some category (or higher category) of "spaces" in which
each space $X$ is equipped with a *fiber*, i.e. a category $C\_X$ of objects
of some type over it.
For example, a space can be a smooth manifold and the fiber
is the category of vector bundles over it;
or a space is an object of the category ... | 34 | https://mathoverflow.net/users/35833 | 22098 | 14,590 |
https://mathoverflow.net/questions/22015 | 19 |
>
> **Definition.** A locally finitely presented morphism of schemes $f\colon X\to Y$ is *smooth* (resp. *unramified*, resp. *étale*) if for any **affine** scheme $T$, any closed subscheme $T\_0$ defined by a square zero ideal $I$, and any morphisms $T\_0\to X$ and $T\to Y$ making the following diagram commute
>
>
... | https://mathoverflow.net/users/1 | Example of a smooth morphism where you can't lift a map from a nilpotent thickening? | Using some of BCnrd's ideas together with a different construction, I'll give a positive answer to Kevin Buzzard's stronger question; i.e., there is a counterexample for *any* non-etale smooth morphism.
Call a morphism $X \to Y$ *wicked smooth* if it is locally of finite presentation and for every (square-zero) nilpo... | 24 | https://mathoverflow.net/users/2757 | 22101 | 14,593 |
https://mathoverflow.net/questions/22089 | 11 | Fix numbers n,k. Is there a closed formula known for the number of k-regular graphs consisting of n edges? I have a method of enumerating k-regular graphs on n edges, and would like to have a number to compare the algorithm against.
| https://mathoverflow.net/users/5002 | Enumeration of Regular Graphs | I think the answer is no, but I would consult the following link:
<http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html>
which contains tables of the sums of the numbers you are interested in. The author is very generous with sharing data that is not posted online.
| 1 | https://mathoverflow.net/users/4542 | 22109 | 14,597 |
https://mathoverflow.net/questions/22105 | 7 | Reading Perelman's preprint(1991) Alexandrov space II now. Got confused about the last section 6.4, which contains an example which indicate that the statement ".... manifold is diffeomorphic to the normal bundle over soul" in Cheeger-Gromoll's Soul Theory won't hold for Alexandrov spaces. I also read BBI's book(A cour... | https://mathoverflow.net/users/3922 | Details of Perelman's example about soul of Alexandrov space | No $X^5$ is not a cone over $CP^2$ and is not compact. The projection has nothing to do with the cone structure. In fact, it's better to forget about the cone structure altogether (until you ask what is the topology of the thing).
The spaces are just quotients of $\mathbb C^3$ and $\mathbb C^2$ by the standard circle... | 9 | https://mathoverflow.net/users/4354 | 22112 | 14,599 |
https://mathoverflow.net/questions/22111 | 24 | Let $U$ be a dense open subscheme of an integral noetherian scheme $X$ and let $E$ be a vector bundle on $U$. Suppose that the complement $Y$ of $U$ has codimension $\textrm{codim}(Y,X) \geq 2$. Let $F$ be a vector bundle on $X$ extending $E$, i.e., $F|\_{U} = E$.
Is any extension of $E$ to $X$ isomorphic to $F$?
| https://mathoverflow.net/users/4333 | Extending vector bundles on a given open subscheme | This is true if $X$ satisfies Serre's condition $S\_2$, i.e. $\mathcal O\_X$ is $S\_2$. Then a vector bundle is $S\_2$ since locally it is isomorphic to $\mathcal O\_X^n$.
More generally, a coherent sheaf $F$ on a Japanese scheme (for example: $X$ is of finite type over a field) which is $S\_2$ has a unique extension... | 37 | https://mathoverflow.net/users/1784 | 22124 | 14,606 |
https://mathoverflow.net/questions/22120 | 11 | More specifically, I was wondering if there are well-known conditions to put on $X$ in order to make $K\_0(X)\simeq K^0(X)$. Wikipedia says they are the same if $X$ is smooth. It seems to me that you get a nice map from the coherent sheaves side to the vector bundle side (the hard direction in my opinion) if you impose... | https://mathoverflow.net/users/14672 | What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes? | Imposing that you can resolve by a length $2$ sequence of vector bundles is too strong. What you want is that there is some $N$ so that you can resolve by a length $N$ sequence of vector bundles. By Hilbert's syzygy theorem, this follows from requiring that the scheme be regular. (Specifically, if the scheme is regular... | 13 | https://mathoverflow.net/users/297 | 22137 | 14,613 |
https://mathoverflow.net/questions/22122 | 9 | Consider a complete $C^\infty$ Riemannian metric on $\mathbb R^2$ of positive sectional curvature.
1. Is the metric embeddable as the boundary of a convex subset of $\mathbb R^3$?
2. Is the embedding unique?
3. Are there generalizations of 1-2 to complete noncompact surfaces of nonnegative sectional curvature?
4. Wh... | https://mathoverflow.net/users/1573 | On Alexandrov embedding theorem | *Is the metric embeddable as the boundary of a convex subset of 3?*
YES, it is a limit case of standard Alexandrov's theorem. Moreover one can choose any embedding of cone at infinity and construct the embedding. This is a [theorem of Olovyanishnikov](http://www.mathnet.ru/php/journal.phtml?wshow=paper&jrnid=sm&paper... | 10 | https://mathoverflow.net/users/1441 | 22143 | 14,615 |
https://mathoverflow.net/questions/22138 | 15 | It's a result of low-dimensional topology that in dimensions 3 and lower, two manifolds are homeomorphic if and only if they are diffeomorphic. Milnor's 7-spheres give nice counterexamples to this result in dimension 7, and exotic $\mathbb{R}^4$'s give nice counterexamples in dimension 4. But I don't know about dimensi... | https://mathoverflow.net/users/3092 | exotic differentiable structures on manifolds in dimensions 5 and 6 | It is false in dimension 5 and 6. Spheres happen to be standard, but some other (compact and closed) manifolds happen to admit different smooth (and PL) structures.
Simple example are tori. For example, $\mathbb T^5$ admits 3 different PL structures that give rise to 3 different differentiable structures. See, e.g., ... | 17 | https://mathoverflow.net/users/2029 | 22147 | 14,618 |
https://mathoverflow.net/questions/22134 | 1 | What is meant by an "ample class" in general? **Motivation:** In the document I am reading, the phrase in question is "fix an ample class $\alpha\in H^1(X,\Omega^1\_X)$." I know what ampleness of a line bundle is. I have checked the only Wikipedia article that could be related (<http://en.wikipedia.org/wiki/Ample_line_... | https://mathoverflow.net/users/5395 | Terminology issue: meaning of 'ample class' ? | Charles' and Pete's answer are (almost) the same: First there is a map
$\mathrm{dlog}\colon \mathcal{O}\_X^\ast \rightarrow \Omega^1\_X$ taking $f$ to
$df/f$ (just to show that it also makes algebraic sense) which indeed induces a
group homomorphism $H^1(X,\mathcal{O}\_X^\ast)\rightarrow H^1(X,\Omega^1\_X)$
giving one ... | 7 | https://mathoverflow.net/users/4008 | 22148 | 14,619 |
https://mathoverflow.net/questions/22142 | 14 | Today I found myself at the Wikipedia page on Vaught's Conjecture,
<http://en.wikipedia.org/wiki/Vaught_conjecture>
and it says that Prof. Knight, of Oxford, "has announced a counterexample" to the conjecture. The phrasing is odd; I interpret it as suggesting that there is some doubt as to whether Prof. Knight atta... | https://mathoverflow.net/users/4367 | Has Vaught's Conjecture Been Solved? | As far as I understand, no, Vaught's Conjecture has not been resolved.
We held a reading seminar on Robin Knight's proposed counter-example closely following each of his drafts and simplified presentations here at Berkeley some years ago and were ultimately convinced that the draft of January 2003 does not contain a ... | 29 | https://mathoverflow.net/users/5147 | 22149 | 14,620 |
https://mathoverflow.net/questions/22065 | 11 |
>
> Does the algebra of continuous
> functions from a compact manifold to
> $\mathbb{C}$ satisfy any specific
> algebraic property?
>
>
>
I'm not sure what kind of algebraic property I expect, but I feel that because of the Gel'fand transform, it may not be unreasonable to expect something. We can drop the co... | https://mathoverflow.net/users/3664 | Algebraic properties of the algebra of continuous functions on a manifold. | I found a reference for a necessary property that might be called algebraic.
Browder [proved a theorem](http://projecteuclid.org/euclid.bams/1183524331) relating the number of generators of a complex commutative Banach algebra to the Čech cohomology with complex coefficients of the maximal ideal space, and as a corol... | 4 | https://mathoverflow.net/users/1119 | 22151 | 14,622 |
https://mathoverflow.net/questions/22154 | 6 | Let $f$ be a function analytic on an open subset $D\subset \mathbb{C}$, and let $\gamma:[0,1] \to D$ be a line segment. $g = f\circ\gamma$ is another curve in the complex plane; is it possible to for $g$ to cross a straight line infinitely often, where the crossing points accumulate towards a point? That is, does there... | https://mathoverflow.net/users/5534 | Can curves induced by analytic maps wiggle infinitely across a line? | The image of $[0,1]$ is compact and so must contain the purported
accumulation point. It makes no loss to assume that $\gamma(t)=t$,
$f(0)=0$ is the accumulation point, and the line in question
is the real axis. Then $f(z)=a\_n z^n+a\_{n+1}z^{n+1}+\cdots$ where
$a\_n$ is nonzero and $n$ is a positive integer.
At this... | 8 | https://mathoverflow.net/users/4213 | 22156 | 14,625 |
https://mathoverflow.net/questions/22161 | 7 | I am looking for the correct technical term in German for the notion of *catenary ring* in commutative algebra.
Does anyone know?
>
> For those who don't know what a catenary ring is but would like to: A Noetherian commutative ring A is called *catenary* if the following codimension formula holds for irreducible ... | https://mathoverflow.net/users/1841 | What is the German translation of "catenary ring"? | It should be "Kettenring", see for example p. 148 in [Brodmann, Algebraische Geometrie](http://books.google.com/books?id=agw3I3IwUDMC&pg=PA256&lpg=PA256&dq=kettenring+primideal&source=bl&ots=M3d7_XqnGb&sig=mFjpisRgp5tnpwnRj0ub45HzTTQ&hl=en&ei=vgTQS8zKNYOQOI7xnZ0P&sa=X&oi=book_result&ct=result&resnum=3&ved=0CBcQ6AEwAg#v... | 13 | https://mathoverflow.net/users/5537 | 22168 | 14,634 |
https://mathoverflow.net/questions/22174 | 29 | When teaching Measure Theory last year, I convinced myself that a finite measure defined on the Borel subsets of a (compact; separable complete?) metric space was automatically regular. I used the [Borel Hierarchy](http://en.wikipedia.org/wiki/Borel_hierarchy) and some transfinite induction. But, typically, I've lost t... | https://mathoverflow.net/users/406 | Regular borel measures on metric spaces | The book *Probability measures on metric spaces* by K. R. Parthasarathy is my standard reference; it contains a large subset of the material in *Convergence of probability measures* by Billingsley, but is much cheaper! Parthasarathy shows that every finite Borel measure on a metric space is regular (p.27), and every fi... | 23 | https://mathoverflow.net/users/1840 | 22177 | 14,639 |
https://mathoverflow.net/questions/22180 | 8 | Is the following statement true?
Given a skew symmetric matrix M, among all of its largest rank sub-matrix, there must be one that is the principal submatrix of M.
| https://mathoverflow.net/users/5543 | Largest rank submatrix of a skew symmetric matrix | Yes. Suppose that you have a general $n \times n$ matrix *M*. The coefficient of $(-x)^k$ in the characteristic polynomial is the sum of all the principal minors of the matrix of size *n-k*. From this it follows that the "rank computed by the principal minors" can differ from the actual rank of the matrix only if the m... | 10 | https://mathoverflow.net/users/4344 | 22183 | 14,643 |
https://mathoverflow.net/questions/22189 | 52 | There are many "strange" functions to choose from and the deeper you get involved with math the more you encounter. I consciously don't mention any for reasons of bias. I am just curious what you consider strange and especially like.
Please also give a reason why you find this function strange and why you like it. Pe... | https://mathoverflow.net/users/1047 | What is your favorite "strange" function? | A **[Brownian motion sample path](http://en.wikipedia.org/wiki/Wiener_process#Some_properties_of_sample_paths)**.
These are about the most bizarrely behaved continuous functions on $\mathbb{R}^+$ that you can think of. They are nowhere differentiable, have unbounded variation, attain local maxima and minima in every ... | 35 | https://mathoverflow.net/users/4832 | 22206 | 14,656 |
https://mathoverflow.net/questions/22118 | 10 | Classical theorems attributed to Levi, Mal'cev, Harish-Chandra for a finite
dimensional Lie algebra over a field of characteristic 0 state that it has a Levi decomposition (semisimple subalgebra plus solvable radical) and that all such semisimple subalgebras (Levi factors) are conjugate in a strong sense: see Jacobson,... | https://mathoverflow.net/users/4231 | Are there "reasonable" criteria for existence/non-existence of Levi factors or their conjugacy in prime characteristic? | [See **Edit** below.]
This isn't really an answer, but I believe it is relevant.
Work geometrically, so $k$ is alg. closed. Let $G$ reductive over $k$, and let
$V$ be a $G$-module (linear representation of $G$ as alg. gp.).
If $\sigma$ is a non-zero class in $H^2(G,V)$, there is a non-split extension
$E\_\sigma$ ... | 3 | https://mathoverflow.net/users/4653 | 22210 | 14,660 |
https://mathoverflow.net/questions/22182 | 3 | I'm a biologist in the process of modeling a fairly simple biological system using a system of ODEs. To verify the simulations, I'm attempting to obtain an analytical steady-state solution that I can check the simulations against. My attempts so far haven't borne fruit, so I thought I'd toss the question out to mathema... | https://mathoverflow.net/users/5544 | Analytical steady-state solution of a complex ODE | As Andrey mentioned, you shouldn't expect an analytical solution since you're dealing with a system of algebraic equation in several variables. (Just to be clear, what we're envisioning here is the system of equations you get by setting all the left-hand-sides to be zero). In your case, I believe you have 5 variables (... | 1 | https://mathoverflow.net/users/353 | 22217 | 14,663 |
https://mathoverflow.net/questions/22125 | 1 | I read that the Euclidean Minimum Spanning Tree (EMST) of a set of points is a subgraph of any Delaunay triangulation. Apparently the easiest/fastest way to obtain the EMST is to find the Deluanay triangulation and then run Prim's algorithm on the resulting edges. However, I have a set of 3D points, and the Delaunay tr... | https://mathoverflow.net/users/5523 | 3D Delaunay Triangulation -> Euclidean Minimum Spanning Tree | Sorry for the confusion - the "wireframe" view of the tetrahedralization of the points was only showing the outside surface (convex hull). Of course all of your comments are correct - the Delaunay triangulation indeed includes ALL points. Then the EMST is a subgraph of it.
All is well - thanks for the confirmations.
... | 0 | https://mathoverflow.net/users/5523 | 22227 | 14,670 |
https://mathoverflow.net/questions/22188 | 44 | I have studied some basic homological algebra. But I can't send to get started on spectral sequences. I find Weibel and McCleary hard to understand.
Are there books or web resources that serve as good first introductions to spectral sequences? Thank you in advance!
| https://mathoverflow.net/users/nan | introductory book on spectral sequences | Many of the references that people have mentioned are very nice, but the brutal truth
is that you have to work **very hard** through some basic examples before it really makes
sense.
Take a complex $K=K^\bullet$ with a two step filtration $F^1\subset F^0=K$, the spectral
sequence contains no more information than is... | 75 | https://mathoverflow.net/users/4144 | 22234 | 14,674 |
https://mathoverflow.net/questions/22232 | 11 | A simplicial set $X$ is a a combinatorial model for a topological space $|X|$, its realization, and conversely every topological space is weakly equivalent to such a realization of a simplicial set. I am wondering, which properties of finite simplicial sets can be effectively (or even theoretically) computed using a co... | https://mathoverflow.net/users/4676 | Which properties of finite simplicial sets can be computed? | For homeomorphism equivalence and homotopy equivalence, the associated problems are recursively unsolvable. This fact dates back to Markov in the 1950s, and relies on the unsolvability of the word problem for finitely presented groups. Apparently it was proved in Markov [1], which is in Russian. That paper has an Engli... | 9 | https://mathoverflow.net/users/5442 | 22245 | 14,681 |
https://mathoverflow.net/questions/22247 | 30 | I don't understand wedge product and Grassmann algebra. However, I heard that these concepts are obvious when you understand the geometrical intuition behind them. Can you give this geometrical meaning or name a book where it is explained?
| https://mathoverflow.net/users/5555 | Geometrical meaning of Grassmann algebra | For a brief explanation of the geometric meaning of exterior product,
interior product of a k-form and l-vector, Hodge dual etc. see my answer here:
[When to pick a basis?](https://mathoverflow.net/questions/4648/when-to-pick-a-basis/4900#4900)
The best reference for this stuff is Bourbaki, Algebra, Chapter 3.
| 10 | https://mathoverflow.net/users/402 | 22254 | 14,685 |
https://mathoverflow.net/questions/22266 | 2 | Someone asked me, and I told them I would try to find out... what is the meaning of this symbol:
B'L or BL'
(I'm not sure if the tick comes before or after the L. It was found on a "nerd clock". The value of this symbol, by the way, is 1.
| https://mathoverflow.net/users/5562 | What is the mathematical meaning of this symbol? | Apparently, it is supposed to be [Legendre's constant](http://en.wikipedia.org/wiki/Legendre%27s_constant), also known as $1$.
| 11 | https://mathoverflow.net/users/1409 | 22268 | 14,692 |
https://mathoverflow.net/questions/21911 | 10 | To ask this question in a (hopefully) more direct way:
Please imagine that I take a freely moving ball in 3-space and create a 'cage' around it by defining a set of impassible coordinates, $S\_c$ (i.e. points in 3-space that no part of the diffusing ball is allowed to overlap). These points reside within the volume, ... | https://mathoverflow.net/users/3248 | When can a freely moving sphere escape from a 'cage' defined by a set of impassible coordinates? | Replace the pins by balls of radius $R\_{ball}$ and the ball by a point. This is a logically equivalent formulation. The question, then, is: given a finite set of balls, $B\_1$, $B\_2$, ...., $B\_k$ in $\mathbb{R}^n$, and a point $x$, how to determine where $x$ is in the unbounded component of $\mathbb{R}^n \setminus \... | 22 | https://mathoverflow.net/users/5563 | 22271 | 14,695 |
https://mathoverflow.net/questions/22255 | 13 | If you have a function $V: L \rightarrow \mathbb{R}$, where $L$ is an infinite dimensional topological vector space, there are multiple notions of differentiability. For $x,u \in L$, $V$ is Gateaux differentiable at $x$ in the direction $u$ if the limit $\underset{t \rightarrow 0}{\lim} \frac{V(x + tu) - V(x)}{t}$ exis... | https://mathoverflow.net/users/2310 | Usefulness of Frechet versus Gateaux differentiability or something in between. | For Lipschitz functions in finite dimensional spaces, Gateaux and Frechet differentiability are the same, but there are huge differences when the domain is infinite dimensional. Lipschitz functions are Gateaux differentiable off a null set when the domain is a separable Banach space and the range has the Radon Nikodym ... | 15 | https://mathoverflow.net/users/2554 | 22273 | 14,696 |
https://mathoverflow.net/questions/22272 | 1 | This may be pretty trivial, but I can't figure it out. Suppose that $S$ is any scheme, and $f\_1:X\_1\to Y\_1$ and $f\_2:X\_2\to Y\_2$ are two morphisms of $S$-schemes, such that the closed image of each one exists. That is, there is a smallest closed subscheme $Z\_i$ of $Y\_i$ over which $f\_i$ factorizes (for $i=1,2$... | https://mathoverflow.net/users/5564 | Closed image of a product of morphisms | This is already false in the affine case.
The closed image of $Spec(B) \to Spec(A)$ is the spectrum of $A/K$, where $K$ is the kernel of $A \to B$. Let $A',B',K$ be analogously defined. The closed image of $Spec(B \otimes\_R B') \to Spec(A \otimes\_R A')$ is the spectrum of $(A \otimes\_R A')/L$, where $L$ is the ker... | 3 | https://mathoverflow.net/users/2841 | 22283 | 14,702 |
https://mathoverflow.net/questions/22289 | 8 | On [page 168](http://books.google.com/books?id=M6DvzoKlcicC&lpg=PP1&dq=mathematical%2520fallacies%2520and%2520paradoxes&pg=PA168#v=onepage&q&f=false) of *Mathematical Fallacies and Paradoxes*, it states that the fact that the series
$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots $
has a sum depends on the A... | https://mathoverflow.net/users/175 | Why does this sum depend on the Axiom of Choice? | It doesn't seem to me that you need any choice principle at all to prove that this series converges. The Alternating Series Test that appears in any elementary calculus books seems to do the job, and doesn't seem to require any amount of AC. If $\Sigma\_n (-1)^n a\_n$ is an alternating series, with $a\_n$ descending to... | 21 | https://mathoverflow.net/users/1946 | 22290 | 14,705 |
https://mathoverflow.net/questions/22195 | 2 | These questions might be elementary for I just started to learn affine Kac-Moody algebra.
It is well known that if we consider finite dimensional Lie algebra, we have the folloing projection:
$R(\lambda)\otimes R(\nu)\rightarrow R(\lambda+\nu)$
where $\lambda$ and $\nu$ are dominant highest weights, $R(\lambda)... | https://mathoverflow.net/users/1851 | Several question on Affine Lie algebra | First question: Peter's (and Emerton's) argument works not only for affine algebras, but for arbitrary Kac-Moody algebras. Decompose your integrable representations into weights, and take the tensor product as the sum of tensor products of weight spaces. It is straightforward to check that the result is still an integr... | 5 | https://mathoverflow.net/users/121 | 22297 | 14,708 |
https://mathoverflow.net/questions/22295 | 4 | What is the difference between:
($\infty,1$) categories - in which have for two objects you have an ($\infty,0$) category of morphisms (i.e. a space of morphisms)
and
categories weakly enriched over spaces - by that I mean categories such that hom(x,y) is always a space and composition is defined only up to (coh... | https://mathoverflow.net/users/1681 | (∞,1) vs Category weakly enriched over spaces | If "space" means the same thing in the two cases, as seems to be implied, then there is no difference, at least not at that level of precision.
There are many different models of $(\infty,1)$-categories. Many of these, like $A\_\infty$-categories, Segal categories, complete Segal spaces, and simplicial categories, tr... | 10 | https://mathoverflow.net/users/49 | 22298 | 14,709 |
https://mathoverflow.net/questions/22302 | 9 | This is a spur of the moment algebraic number theory question prompted by a side remark I made in a course I'm teaching:
Let $K$ be a number field. The (Hilbert) **class field tower** of $K$ is the sequence defined by $K^0 = K$ and for all $n \geq 0$, $K^{n+1}$ is the Hilbert class field of $K^n$. Put $K^{\infty} = \... | https://mathoverflow.net/users/1149 | Algorithm for the class field tower problem? | Not in the slightest! The answer is not even known for quadratic imaginary number fields. In fact, the *only* known way to show that the Hilbert class field tower of a number field is infinite is to invoke one of a variety of different forms of Golod-Shafarevich, and I don't think it's even seriously conjectured (more ... | 13 | https://mathoverflow.net/users/35575 | 22306 | 14,714 |
https://mathoverflow.net/questions/22309 | 4 | Some of the fundamental results in analysis (inverse function theorem, existence and uniqueness of solutions to ODEs) have slick proofs using the idea of a contraction. So, it seems plausible to me that one might be motivated to study a "contraction space":
I'll define a contraction space as a metric space $(X,d)$ su... | https://mathoverflow.net/users/2363 | Is a "contraction space" always complete? | There is a "contraction space" which is not complete. For example, consider a metric $d$ on $[1,+\infty)$ such that for $x,y\in[n,n+1]$ where $n\in\mathbb N$ one has $d(x,y)=2^{-n}|x-y|^{1/n}$ (other distances are defined by gluing the segments together). The completion is obtained by adding one point at $+\infty$.
B... | 9 | https://mathoverflow.net/users/4354 | 22314 | 14,720 |
https://mathoverflow.net/questions/22319 | 5 | If $f:X \to Y$ is a flat and proper surjective morphism between smooth schemes over an algebraically closed field, and $f$ has connected fibers, does it imply that
$$f`\_\*\mathcal O\_X = \mathcal O\_Y?$$
| https://mathoverflow.net/users/5464 | Is the direct image of the structure sheaf on X isomorphic to the structure sheaf on Y when X->Y is flat and proper with connected fibers between smooth schemes over an algebraically closed field? | This follows from Zariski's main theorem if the characteristic is zero and it is false in positive characteristics: consider the the morphism $\mathbb{A}^1 \to \mathbb{A}^1$ given by $x \mapsto x^p$ where $p$ is the characteristic. The statement would also be true in char p if you assume that the general fibre is reduc... | 11 | https://mathoverflow.net/users/519 | 22333 | 14,731 |
https://mathoverflow.net/questions/22327 | 9 | We say that a subset $X$ of $\mathbb{R}$ is midpoint convex if for any two points $a,b\in X$ the midpoint $\frac{a+b}{2}$ also lies in $X$.
My question is: is it possible to partition $\mathbb{R}$ into two midpoint convex sets in a non-trivial way?
(trivial way is $\mathbb{R}=(-\infty,a]\cup(a,+\infty)$ or $\math... | https://mathoverflow.net/users/5572 | Partition of R into midpoint convex sets | Yes if you assume AC:
With AC let $\{v\_\alpha\}$ be a $\mathbb{Q}$-basis for $\mathbb{R}$ then the following two sets satisfies your property:
$A = \{q\_1v\_{\alpha\_1}+\cdots+q\_nv\_{\alpha\_n} \mid q\_i \in \mathbb{Q} , \sum q\_i \geq 0 \}$
and
$B = \{q\_1v\_{\alpha\_1}+\cdots+q\_nv\_{\alpha\_n} \mid q\_i \... | 11 | https://mathoverflow.net/users/4500 | 22338 | 14,734 |
https://mathoverflow.net/questions/22320 | 4 | As the title says I would like to know if $K\_1(k)=k^\*$ uniquely determines a field $k$.
For finite fields this is clearly the case, but I suspect it is not ture in general. However I guess cooking up a counterexample is not so easy.
| https://mathoverflow.net/users/2837 | Is a field uniquely determined by its multiplicative group/how much knows K_1 about fields? | Let $K$ and $L$ be two algebraically closed fields of characteristic $0$. Then $K^{\times} \cong L^{\times}$ iff $K$ and $L$ have the same cardinality.
The forward direction is clear. Conversely, if $K$ is algebraically closed, then consider
the short exact sequence
$1 \rightarrow K^{\times}[\operatorname{tors}] \... | 5 | https://mathoverflow.net/users/1149 | 22341 | 14,736 |
https://mathoverflow.net/questions/22281 | 5 | I would like to express the integral of the absolute value of a real-valued function $f$ (over a finite interval) in terms of the Fourier coefficients of $f$. Failing that, I would like to know of any constraints or statistical correlations (in a sense explained in the motivation) relating these quantities.
Motivatio... | https://mathoverflow.net/users/3291 | Can I relate the L1 norm of a function to its Fourier expansion? | This sounds difficult. For instance, a long-standing open problem of Littlewood used to be this. Let A be a set of integers of size n, and let f be the characteristic function of A. How small (up to a constant) can the sum of the absolute values of the Fourier coefficients of f be? The conjecture was that the smallest ... | 8 | https://mathoverflow.net/users/1459 | 22367 | 14,755 |
https://mathoverflow.net/questions/22364 | 0 | In comparing the norm of two operators, I come across the following problem.
Let $S\in M\_{n}(\mathbb{R})$ be a symmetric matrix. $D\_1=diag(\alpha\_1,\cdots,\alpha\_n)$, $D\_2=diag(\beta\_1,\cdots,\beta\_n)$, with $\alpha\_1\ge\cdots\ge\alpha\_n\ge0, ~\beta\_1\ge\cdots\ge\beta\_n\ge 0$. Is it true that $Tr[(D\_1S^2D... | https://mathoverflow.net/users/3818 | Is this trace inequality true? | I just wrote down some random 2x2 example, and it had the wrong sense for the inequality.
A = [2,1;1,0], D1=D2=diag(2,1) .
| 8 | https://mathoverflow.net/users/454 | 22376 | 14,760 |
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